As discussed in the note on Huygens’ Principle, if we separate the
solution y(r,t) of the usual wave
equation in n-dimensional space (with one time dimension) into a time component
and a spatial component, we have y(r,t)
= f(r)g(t), and the spatial and temporal components satisfy the individual equations

where k is a constant with units of 1/distance and w is a constant with units of 1/time. Thus
the temporal component satisfies the simple harmonic equation, with a general
solution of the form g(t) = g1 eiwt + g2 e-iwt
where g1 and g2 are arbitrary constants. In just one
spatial dimension (n = 1) the spatial equation also reduces to the simple
harmonic equation, with the general solution f(r) = f1 eikr
+ f2 e-ikr for
constants f1 and f2. Combining these, we get the wave
function

Thus the solution is a sum of functions of the quantities
kr + wt and kr – wt. If we require that f and g are
real-valued, then g(t) = g1 cos(wt), f(r) = f1 cos(kr),
and

More generally, we can verify by direct substitution that
the one-dimensional wave equation is satisfied by any function of the form

where A(x) and B(x) are quite arbitrary functions. In
effect, Huygens’ Principle can be read directly from this equation, since it
implies that a pulse disturbance propagates sharply at a constant speed. We
also know that with three spatial dimensions (n = 3) the general spatial
solution is f(r) = cos(kr)/r, and again Huygens’ principle applies. We
asserted that a similar result obtains for any odd number of spatial
dimensions. To show this more explicitly, recall that spatial equation is

We assume f(r) has a power series expansion

where c0 is the first non-zero coefficient and q is
non-negative (to ensure that f(r) is finite at r = 0). Inserting this series
and its derivatives into equation (1) and setting the coefficient of each
power of r to zero, we get the conditions

and so on. Since c0 is stipulated to be
non-zero, and since we are requiring q to be non-negative, the first of these
conditions implies q = 0, and so the second implies c1 = 0. The
remaining conditions give cj + 2 as a multiple of cj,
so it follows that cj = 0 for all odd j. The coefficients with
even indices are then given by

With n = 3 the spatial component of the wave function is
therefore

On the other hand, in a five-dimensional space, we have n
= 5, and the spatial part of the wave function is

Putting s = kr and letting a(s)
denote the expression inside the last square brackets, we see that

Multiplying through by s and integrating, we find that a(s) = sin(s) – s cos(s), so we have

The same general approach allows us to determine the
closed-form expression for fn(r) for any odd number of dimensions
n. For example, in seven spatial dimensions (n = 7) we have the series

Again putting s = kr and letting a(s) denote the expression inside the last square brackets, we
see that

Multiplying and integrating twice gives

Therefore, the spatial part of the spherical solution of
the wave equation in seven space dimensions is

The same approach leads to the solution in nine space
dimensions

and so on. In general, the spatial part of the spherical
wave solution in n dimensions has the form

where the constant Kn equals (n-2)(n-4)…(1),
and the expressions An(kr) and Bn(kr) denote “even” polynomials
in kr. These polynomials for the first several odd values of n are listed
below.

In general, for odd n greater than 1, these polynomials
can be expressed as

Interestingly, referring to the article
on proving that p is irrational, we
see that the spatial functions fn(r) for odd integers n greater
than 1 can also be expressed (up to a constant factor) as a simple integral

The figure below shows fn(r) for spaces of one,
three, and five dimensions.

Of course, these are just the elementary solutions. We can
use linear combinations of solutions of this form to generate arbitrary wave
forms.

As an aside, although the case of negative n presumably has
no physical significance (negative dimensions?), we note that a similar
approach enables us to solve equation (1) for these cases as well. If n is
even, the coefficients are undefined (because they involve a division by
zero), but we get well-defined functions for odd n. The solutions for the
first couple of odd negative values of n are

These solutions increase to infinity as r increases, unlike
the solutions for positive n, which drop to zero as r increases.

Returning to positive dimensions, there’s an interesting
relationship between the “basis functions” of the general spherical wave
solutions in successive odd dimensions. The wave equation in spherical
coordinates in n-dimensional space can be written in terms of the
differential operator

A function y(r,t)
is a solution of the n-dimensional wave equation if and only if

Knowing the general form of the spatial part fn(r),
and the simple temporal part for all n, we can write a basic combined
space-time solution for odd n as

where An(r) and Bn(r) are the
polynomials defined previously. By direct substitution it can be verified
that successive odd basis solutions are related by

This is a remarkable fact, signifying that a basis solution
for n space dimensions not only satisfies the wave equation everywhere in
n-dimensional space, it also satisfies the wave equation in spaces of every
odd number of dimensions at every radius and time for which the basis
solution for n+2 dimensions vanishes. Hence there is an infinite sequence of
expanding (or contracting) discrete shells on which any basis solution
satisfies the wave equation for spaces of all odd dimensions.

In the special case m = 1 the spherical wave operator
reduces to

and we have the recurrence

We refer to the fn
as basis solutions, because it’s easy to see that if fn(r,t) is a solution, then so
is kfn(jr,jt) for any
constants k and j. For example, each of the following expressions are
solutions for n = 5:

and so on. Any linear combination of these solutions is
also a solution, so by means analogous to Fourier series we can construct
arbitrary functions of the form

and similarly for Gs(r-t) and Gc(r-t).
The analogous functions can be constructed from the basis solutions for any
odd n, so the spherical wave equation in n space dimensions is satisfied by

for arbitrary functions F and G. Notice that the greater of
the degree of An and the degree of rBn is (n-3)/2, so the lowest inverse power of r is
(n-2) – (n-3)/2 = (n-1)/2. This
is consistent with the fact that the energy of a wave is proportional to the
square of the amplitude, so the energy per unit “area” of the spherical wave
drops in inverse proportion to rn-1, which is the dimension of the surface of a sphere in
n-dimensional space. It’s interesting that, for odd space dimensions greater
than 3, the amplitude contains formal terms that drop in inverse proportion
to higher powers of r as well. This is in a sense misleading, because the
amplitude (and hence the energy per unit area) is continuously changing as
the wave propagates to greater values of r, and at the same time the value of
the wave function is changing with phase. The wave function is not actually
periodic, so the correspondence between energy and “amplitude” contains an
ambiguity, which manifests itself in the higher-order terms in higher
dimensions.